Learn Arabic (Modern Standard): هاجم الخصم، ولا مفر من المحكمة اليوم. - The opponent attacked, and there is no escape from the court today.

This is the classic behavior of لا of absolute negation (لا النافية للجنس), which you are learning now. When it negates an entire category to mean 'absolutely no escape', the noun immediately following it takes a fatha but drops the tanween, making it مفر (mafarra) instead of مفرا (mafarran).

Is هاجم a noun or a verb here? It looks like the word for 'attacker'.

It is a Form III past-tense verb meaning 'he attacked'. Because it follows the pattern 'faa'ala', its unvoweled spelling looks identical to an active participle (a noun or adjective meaning 'attacker'). The sentence structure tells us it is a verb, because it is immediately followed by its subject, الخصم (the opponent).

What is the difference between saying لا مفر and ليس هناك مفر?

Using the absolute negation لا is much stronger and more literary. While ليس هناك مفر simply states 'there is no escape', لا مفر categorically denies the very existence of any escape whatsoever.

How is the word مفر built?

It comes from the root ف-ر-ر (f-r-r), which means to flee or run away. The prefix مـ (ma-) is often added to roots to create a noun indicating a place or an action. Here, it literally translates to 'a place of fleeing' or 'an escape'.